Multinomial Logistic Regression | Mplus Data Analysis Examples

Multinomial logistic regression is used to model nominal
outcome variables, in which the log odds of the outcomes are modeled as a linear
combination of the predictor variables.

Please note: The purpose of this page is to show how to use various data analysis commands.
It does not cover all aspects of the research process which researchers are expected to do. In
particular, it does not cover data cleaning and checking, verification of assumptions, model
diagnostics and potential follow-up analyses.

Examples of multinomial logistic regression

Example 1. People’s occupational choices might be influenced
by their parents’ occupations and their own education level. We can study the
relationship of one’s occupation choice with education level and father’s
occupation. The occupational choices will be the outcome variable which
consists of categories of occupations.

Example 2. A biologist may be
interested in food choices that alligators make. Adult alligators might have
different preferences from young ones. The outcome variable here will be the
types of food, and the predictor variables might be size of the alligators
and other environmental variables.

Example 3. Entering high school students make program choices among general program,
vocational program and academic program. Their choice might be modeled using
their writing score and their social economic status.

Description of the data

For our data analysis example, we will expand our third example with a
hypothetical data set. The data set contains variables on 200 students. The outcome variable is
prog, program type, where program type 1 is general, type 2 is academic,
and type 3 is vocational. The predictor variables are social economic status,
ses, a three-level categorical variable and writing score, write, a continuous variable. Let’s start with getting some descriptive statistics of the variables of interest. You can download the
data set here.

Multiple logistic regression analyses, one for each pair of outcomes:
One problem with this approach is that each analysis is potentially run on a different
sample. The other problem is that without constraining the logistic models,
we can end up with the probability of choosing all possible outcome categories
greater than 1.

Collapsing number of categories to two and then doing a logistic regression: This approach
suffers from loss of information and changes the original research questions to
very different ones.

Ordinal logistic regression: If the outcome variable is truly ordered
and if it also satisfies the assumption of proportional
odds, then switching to ordinal logistic regression will make the model more
parsimonious.

Alternative-specific multinomial probit regression: allows
different error structures therefore allows to relax the independence of
irrelevant alternatives (IIA, see below "Things to Consider") assumption.
This requires that the data structure be choice-specific.

Nested logit model: also relaxes the IIA assumption, also
requires the data structure be choice-specific.

Multinomial logistic regression

Below we show how to regress prog on ses and write in a
multinomial logit model in Mplus. We specify that the dependent variable,
prog, is an unordered categorical variable using the Nominal option. Mplus
will not automatically dummy-code categorical variables for you, so in order to
get separate coefficients for ses groups 1 and 2 relative to ses group 3, we
must create dummy variables using the Define command. We include our newly
created dummy variables, ses1 and ses2, in both the Usevariables option and the
Model command. In the multinomial logit model, one
outcome group is used as the "reference group" (also called a base category), and the
coefficients for all other outcome groups describe how the independent variables
are related to the probability of being in that outcome group versus the reference
group. Mplus automatically uses the last
category of the dependent variable as the base category or comparison group,
which in this case is the vocational category.
Looking at the syntax below, in the model statement we have entered "prog#1
prog#2 on ses1 ses2 write." Mplus uses a variable name followed by a pound sign
and a number to refer to the categories of the nominal dependent variable, except the final category,
which is the reference group and cannot be referred to in the model statement
(if you try, Mplus will issue an error message). Thus the
line included in our model statement indicates that we want to regress both
levels of prog on ses(as dummy variables) and write.
Additionally, by default for multinomial logistic regression, Mplus calculates
robust standard errors.

In the output above we see the final log likelihood (-179.982), which can be used
in comparisons of nested models.

Under the heading “Information Criteria” we see the Akaike and Bayesian information
criterion values. Both the AIC and the BIC are measures of fit with some correction
for the complexity of the model, but the BIC has a stronger correction for parsimony.
In both cases, lower values indicate better fit of the model.

The output above has two parts, labeled with the categories of the
outcome variable prog. They correspond to the two equations below:

A one-unit increase in the variable write is associated with a
0.056 increase in the relative log odds of being in general program vs.
vocational program .

A one-unit increase in the variable write is associated with a
0.114 decrease in the relative log odds of being in academic program vs.
vocational program.

The relative log odds of being in general program vs. in vocational program will
decrease by 0.645 if moving from the highest level of ses (ses==3) to the
middle level of ses (ses==2).

The ratio of the probability of choosing one outcome category over the
probability of choosing the baseline category is often referred to as relative risk
(and it is also sometimes referred to as odds as we have just used to described the
regression parameters above). Relative risk can be obtained by
exponentiating the linear equations above, yielding regression coefficients that
are relative risk ratios for a unit change in the predictor variable. These
relative risk ratios can be found in the Logistic Regression Odds Ratio Results
section of the output.

The relative risk ratio for a one-unit increase in the variable write
is 1.057 (exp(0.056) from the Model Results output) for being in general program vs.
vocational program.

The relative risk ratio switching from ses = 3 to 1 is
1.197(exp(0.180) from the Model Results output) for being
in general program vs. vocational program. In other words, the expected risk
of staying in the general program is higher for subjects who are low in
ses.

Things to consider

The Independence of Irrelevant Alternatives (IIA) assumption: roughly,
the IIA assumption means that adding or deleting alternative outcome
categories does not affect the odds among the remaining outcomes.

Diagnostics and model fit: unlike logistic regression where there are
many statistics for performing model diagnostics, it is not as
straightforward to do diagnostics with multinomial logistic regression
models.
For the purpose of detecting outliers or influential data points, one can
run separate logit models and use the diagnostics tools on each model.

Pseudo-R-Squared: the R-squared offered in the output is basically the
change in terms of log-likelihood from the intercept-only model to the
current model. It does not convey the same information as the R-square for
linear regression, even though it is still "the higher, the better".

Sample size: multinomial regression uses a maximum likelihood estimation
method, it requires a large sample size. It also uses multiple
equations. This implies that it requires an even larger sample size than ordinal or
binary logistic regression.

Complete or quasi-complete separation: Complete separation implies that
the outcome variable separates a predictor variable completely, leading to
perfect prediction by the predictor variable. Perfect prediction means that only one value of a predictor
variable is associated with only one value of the response variable. You can do a two-way tabulation of the outcome
variable with the problematic variable to look for separation, and if
detected, rerun the model
without the problematic variable.

Empty cells or small cells: You should check for empty or small
cells by doing a cross-tabulation between categorical predictors and
the outcome variable. If a cell has very few cases (a small cell), the
model may become unstable or it might not even run at all.